1% Smaller Every Time

Since I found the McElroys and their several unconscionably good podcast properties a few years ago, I’ve been trudging through the backlog in hopes of one day getting caught up. I’m up to date on all of the McElroy properties I consume, except one: My Brother, My Brother, and Me, that treasure trove of good advice.

I recently listened to episode 177, Juicy Crust, released 26 November 2013. Towards the end of the show, the boys consider a question regarding the existence of double-ghosts—that is, people who have died once as humans and also a second time as ghosts. Naturally, if such a thing were to exist, there would be all sorts of supernatural implications, and who better than three dudes from West Virginia to work them out.

The Question Within the Question

At 41:55, Travis offers a side-question to the group:

What if, each time the ghost died and came back, they were just 1% smaller […] but also 1% denser?

That question gave me pause. Putting numbers to the issue, after all, would make it easier to predict the consequences.

Unfortunately, there’s a problem. Travis’ question leaves open some possibility for interpretation. Everyone can agree on how to make a ghost more dense—you just add enough ghost mass to the ghost volume to make the density 1% more than when you started—but how exactly do you make the ghost smaller? You see, you could make the ghost smaller by chopping off a little bit of ghost meat, taking the ghost mass with it (and probably angering the ghost), or you could just pack all the ghost mass into a slightly smaller volume, thus increasing the ghost density even more.

Travis goes on to say something about dark matter, which I tend to think means he’s thinking that the mass of the ghost will increase to celestial levels and become a hilarious inconvenience. For this reason—and because the results of the first interpretation are sort of boring, since the ghost mass would sort of just decay into nothingness—we will consider the second interpretation above as the canonical one.

Let’s get started.

The Math

Density is just the ratio of mass to volume.

We’ll assume that ghosts are of uniform density. For all I know, they are.

Let’s use the variable $n$ to count the number of times the ghost has died and been re-ghosted. That means we’re looking for some function $\rho(n)$ that tells us how much mass the ghost has after $n$ death-ghost cycles. We’ll also use $m_{i}$, $V_{i}$, and $\rho_{i}$ as symbols for the mass, volume, and density of the ghost at the beginning of our horrible, horrible thought experiment.

We want the ghost density to increase by 1% after each cycle, so let’s add that to our function.

We also want ghost volume to decrease by 1%, so we’ll throw that in.1

This can be rewritten more simply as the following.2

This function is scary. And not, like, ooooo ghooooossts scary, although obviously that, too. No, this function is scary because it grows. You see, some functions like to explode, but functions like this one start out slow. Over time, they gain speed, and pretty soon things get cosmic.

After one cycle, the ghost has about 2% more mass than before. This makes sense, since we added 1% more density and reduced the volume by 1%. It’s not quite exactly 2% because of how percentages work, but it’s close enough for now. If this ghost weighed as much as the average American, they’d be about 4 pounds (2 kg) heavier. Since their volume is decreasing, they’d also be about two inches (5 cm) shorter.

By ten cycles, the ghost is now 21% heavier than when we started. They’re 40 pounds (18 kg) heavier, but they’re also about a foot and a half (45 cm) shorter.

After 100 cycles, things start to get weird. Our ghost is now 4.4 times heavier and 2.7 times smaller than when we started. Two of them put together would be as heavy as a dairy cow, but they’re only about a foot (30 cm) across. They are now more dense than human bone, except their whole ghostly body is that density.

Once we pass 1000 cycles, we leave the realm of reality (even where ghosts are concerned). Ghost mass is now over 44,000 times greater than it was at the beginning, but ghost volume is now only 0.004% what it was. The ghost is now denser than any material, known or theoretical, and may be more dense than anything can possibly be. The ghost became a black hole a long time ago, but repeated deaths and re-ghostings (can black holes die?) have resulted in this unholy concoction of our imagination.

So there you have it.

From the parameters put forth by the middlest brother, we can conclude that such ghosts would only have a few hundred cycles available to them before they became unphysical insults to the universe. If the initial conditions were reduced (less mass, less density, or more volume), then the ghost might have a few more cycles to spare. In any case, its days (and all of our days, too) would be numbered.

  1. Note that we have to subtract a $\rho_{i}$ because the last two terms work together to provide the additional density contributed by the volume decrease, not the new density after that operation on its own. This bit took a while for me to figure out, and I hope it’s right. 

  2. Recall that $\rho = \frac{m}{V}$, so $\frac{m_{i}}{V_{i}}$ can be reduced to simply $\rho_{i}$ and factored out with the others.